arXiv subjects
Publications and source records attributed to Andrei Kapaev.
We describe the close connection between the linear system for the sixth Painlevé equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.
The general and special repo rates are related with the prices of the European call- and American put-options. The evaluation takes into account specific business models of the parties in the repo agreement and the law restrictions. Using the repo-option relation, an alternative to the Black-Scholes method of option pricing is presented. The empirical data on the general and special repo rates are explained.
We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2) it does not possesses the Painlevé property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data.
The 0-instanton solution of Painlevé I is a sequence $(u_{n,0})$ of complex numbers which appears universally in many enumerative problems in algebraic geometry, graph theory, matrix models and 2-dimensional quantum gravity. The asymptotics of the 0-instanton $(u_{n,0})$ for large $n$ were obtained by the third author using the Riemann-Hilbert approach. For $k=0,1,2,...$, the $k$-instanton solution of Painlevé I is a doubly-indexed sequence $(u_{n,k})$ of complex numbers that satisfies an explicit quadratic non-linear recursion relation. The goal of the paper is three-fold: (a) to compute the asymptotics of the 1-instanton sequence $(u_{n,1})$ to all orders in $1/n$ by using the Riemann-Hilbert method, (b) to present formulas for the asymptotics of $(u_{n,k})$ for fixed $k$ and to all orders in $1/n$ using resurgent analysis, and (c) to confirm numerically the predictions of resurgent analysis. We point out that the instanton solutions display a new type of Stokes behavior, induced from the tritronquée Painlevé transcendents, and which we call the induced Stokes phenomenon. The asymptotics of the 2-instanton and beyond exhibits new phenomena not seen in 0 and 1-instantons, and their enumerative context is at present unknown.